At end of 6th chapter of Melakhim II, there is a siege by Aram around Shomron & there is a famine in Shomron, but a miracle occurs and the army of Aram hears sounds of armies come towards them so they flee. The inhabitants of Shomron get the spoils from Aram and the famine is over. It is in this context that we have the famous story of the metzoraim who discovered the army of Aram. Rav Aaron Soloveitchik used this story as a precedent that sometimes God can bring about salvation even by the hands of people who were not of the highest spiritual caliber. Therefore, he asserted that the creation of the state of Israel can be considered an act of salvation by God even though it was founded by a secular government.
In the 8th chapter, Elisha tells Hazael that he will be the king of Aram, but he will make Bnei Yisrael suffer so Elisha weeps over this. Meanwhile, Yehoram becomes king of Yehuda when Yehoshafat dies and he was wicked and when he dies Achazyahu reigns in his stead and he was also wicked.
In the 9th and 10th chapters, Yehu is told by the servant of Elisha that he will be king and Yehu wipes out the house of Achav, including Yehoram, Izevel and Achav's 70 children. Unfortunately, Yehu did not observe the word of God either as he quickly fell to sin.
It is interesting that Elisha sends a messenger to anoint Yehu and that Elisha doesn't anoint Yehu himself. Why is Elisha essentially uninvolved in the rebellion of Yehu? It seems that there is a strong connection between the actions of Yehu and the prophecies of Eliyahu. In fact, God commanded Eliyahu to anoint Yehu as the king of Israel during the reign of Achav. Eliyahu didn't fulfill this command, so it seems that this command was passed on to Elisha.
The prophecy that Elisha's servant told to Yehu is essentially what Eliyahu told Achav would happen, but because Achav repented, the punishment was only carried out in the days of his children. But Yehu understands that the prophecy of Eliyahu was the basis for his rebellion. He eliminates Baal worship, and it seems that the reason for the destruction of Achav's kingdom was his fervent worship of Baal. As such, destruction of Achav's kingdom also entails destruction of Baal. One might argue that Yehu, in wiping out Achav's kingdom, acted similar to Eliyahu killing the Baal prophets at Har HaCarmel. In both instances, there was no explicit commandment by God to do so, but it was the right thing to rid the country of these corrupt idolaters. One can conclude that that which Eliyahu prophesized became a political reality because of Yehu and therefore, the behavior of Yehu is not a fulfillment of Elisha's wishes but really a fulfillment of Eliyahu's wishes.
But why didn't Elisha anoint Yehu? Why did he send a messenger instead? R. Elchanan Samet suggests that Elisha was not the zealot that Eliyahu was so he could not personally fulfill this task despite the fact that he succeeded Eliyahu. Elisha could report prophecies that he had witnessed, but he himself had to send someone else to tell one Jew to kill other Jews, even if it was the will of God.